1. Introduction

Quasi-phase matching (QPM) devices based on ferroelectric crystals are widely used in applications such as highly efficient optical frequency converters and parametric optical amplifiers/oscillators. For the optimal performance of a QPM device, the poled domain structure must ensure good fidelity to the designed grating structure. In the standard electric field poling process using e-beam-defined photo-masks, high-quality QPM devices with sharp domain boundaries can be obtained from various ferroelectric crystals. Although such a photolithographic process assures stable periodicity in the QPM grating, randomness in the domain boundary locations is inevitable in ferroelectric QPM devices [1]. Such randomness, negligible in conventional gratings, does occur during high-voltage ferroelectric domain reversal, due to non-uniform expansion of the reversed ferroelectric domain regions. The statistical departure of the domain boundaries from the ideal locations, called the random duty-cycle error (RDE) by Pelc et al., can lead to decreased efficiency as well as undesired function of the devices such as parasitic harmonic generation [2]. In a reasonably good QPM device, the decrease in the conversion efficiency is not significant, but the non-phase-matched parasitic generation can be detrimental to photon-level frequency conversion applications [2–5].

Although the RDE of a QPM device can be directly evaluated by measuring the widths of poled and unpoled domains using a microscope, the measurement of an entire QPM channel (typically made of ~1,000 domains or more) is time consuming, which is not desirable for either device makers or users. Thus, several indirect methods for poling quality evaluation have been developed [2, 6–9]. Among them, wavelength-tunable second-harmonic generation (SHG) is representative, and gives a direct estimation of the performance as other types of frequency conversion devices do [1]. The relationship between the RDE and the efficiency of the SHG pedestal has been established in [2]. However, most of these methods are more sophisticated than the direct microscopic measurement, and sometimes not practical. The tunable SHG experiment is not easy either; it requires tunable narrow-linewidth sources, and the tuning range is significantly limited even with two tunable lasers in sequence [2]. Thus, it allows the measurement of the parasitic generation only near the first order QPM peak, but cannot give experimental estimation of the amount far from the QPM peaks.

In this work, instead of a tunable SHG experiment, we measured the far-field diffraction pattern from the QPM grating, which is mathematically equivalent to the SHG tuning curve. The mathematical equivalence results from the fact that both of them are Fourier transforms of the grating structure, when the SHG spectrum is calculated under the negligible depletion limit, and has been experimentally proven valid [10,11]. The same RDE information was obtained from a much simpler diffraction experiment with a low-power laser. Furthermore, the pure background noise far from the orders could be measured, overcoming the limited tunable range in the tunable SHG experiment.

In our previous works [10,11], a small beam (focused spot) was used to detect the first order and the second order diffraction intensities from a QPM grating, and their ratio gave the local duty ratio. Because the statistics of the duty ratio was obtained by scanning the spot throughout the QPM channel, the RDE was somewhat underestimated due to the averaging effect within the focused spot illuminating a few periods. In the present work, we used an expanded, collimated beam to cover the whole channel of the QPM channel. The RDE was obtained directly from the diffraction pattern without scanning a focused spot, and the ‘pre-averaging’ problem was also overcome.

2. Theory

Here, we introduce a realistic model for the random domain structure fabricated by the standard electric field poling method, and calculate the far-field diffraction pattern as a function of the standard deviation (STD) of the duty ratio. Since the standard electric field poling method employs a photo-mask, the QPM period is kept uniform throughout the channel. Our model is similar to the one described in [2], but different in that the average duty ratio is not necessarily 1/2, as explained below. First, we assumed that the electrode locations are perfectly periodic, and that the poled domain created under each electrode expands randomly to both right and left sides as schematically described in Fig. 1. We considered the random locations of the domain boundaries at both sides. In Fig. 1, c_k = Λ (k−N/2) (k = 1, 2, …, N) represents the exact locations of the center of the k-th electrode, where Λ is the QPM period and N is the number of periods. w_k^l and w_k^r are the distances of the left and right domain boundaries from the center, respectively, whose distributions are assumed to be Gaussian with the same average and STD. Then, $x_k^r=c_k+w_k^r/2$and$x_k^l=c_k-w_k^l/2$are the coordinates of the right and the left boundaries of the poled domain, respectively.

 

Fig. 1 Schematic diagram of random domain model. (P_s: spontaneous polarization)

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After poling, the top electrodes are removed, and the bottom surface is slightly etched to reveal a surface-relief phase grating structure. For a plane light wave illuminating the random phase grating at normal incidence, we can write the transmission function t(x) as follows:

If the grating structure is ideal (σ = 0), f(ξ) = 1 and the first term of Eq. (3) vanishes. Then, the diffraction intensity pattern represented by the second term is that of a conventional binary phase grating exhibiting pronounced narrow orders (X = 1, 2, 3 …), and small tails around them. For a grating with disorder, however, the strength of each diffraction order is decreased by the factor f (ξ), and the first term adds a background noise to the tails of the second term. For a QPM device with small disorder (ε ≡ σ/Λ << 1), f(ξ) ≈1 – (πξσ)², the diffraction efficiency of each order is not significantly affected by σ, while the noise term is roughly proportional to ε²/N, exhibiting a flat background in the spatial frequency spectrum [12].

In practice, it is difficult to measure the absolute noise intensity for a given input intensity. Instead, it is much easier to measure the relative intensity of the noise with reference to one of the diffraction orders, within the same diffraction pattern. We measured the noise intensity (I_n) at the midpoint (X = 1.5) between the first order (X = 1) and the second order (X = 2), and compared it with the first order intensity (I₁). For a grating with small disorder (σ/Λ << 1), the relative noise intensity is obtained directly from Eq. (3),

We also point out that the SHG tuning curve is obtained through the same procedure simply by replacing Q = e^iϕ − 1 with Q = −2 in Eq. (3), showing the mathematical equivalence between the SHG tuning curve and the diffraction pattern.

3. Experiment

For the demonstration of our method, we periodically poled a z-cut congruent LiNbO₃ crystal plate by the standard electric field poling method using a photo-mask. In order to produce a phase grating for the diffraction experiment, the original negative z-face of the periodically poled LiNbO₃ (PPLN) sample was slightly etched with an acid solution, giving a uniform depth of 31 ± 2 nm throughout the channel. For our sample, we observed that the domain boundaries were almost parallel to the ferroelectric axis, confirming that the etched surface grating structure is an accurate projection of the QPM structure. The grating channel was about 10 mm long and 3 mm wide, with a period of 21.2 μm.

We used a He-Ne laser (633 nm, 2.5 mW) as a light source for the diffraction experiment. The laser beam was expanded and collimated by a pair of cylindrical lenses, illuminating the PPLN grating channel at normal incidence. The grating region of 10 mm was uniformly illuminated by limiting the input beam with a pair of blockers placed directly in front of the sample. We used a convex lens with a focal length of 1 m just behind the sample, in order to obtain the far-field diffraction pattern at the focal plane. A CCD-camera was used to record a rough diffraction pattern, shown as the inset of Fig. 2(a). For a quantitative measurement of the intensity, however, a Si-photodetector replaced the CCD-camera due to its poor linearity. The Si-photodetector with an attached slit was scanned along the diffraction pattern on the Fourier plane. A slit width of 0.05 mm was used to resolve the intensity profiles of the narrow orders, while it was increased to 3.0 mm in the noise region. A greater slit width not only smoothens the background noise, but also helps enhance detection sensitivity by increasing the amount of light received by the detector. For comparison, we performed the same experiment on a reference grating (period 26 μm), which was made on a glass plate as a thin photoresist grating. We took microscopic pictures of both the PPLN and the reference grating, and measured the domain widths for verifying the validity of the suggested diffraction method.

 

Fig. 2 Far-field diffraction patterns for PPLN (a) and reference grating (b). Inset of (a) is a CCD-image. Two bright spots indicate the first and second orders (look larger due to saturation of CCD). The region between them was enhanced to show small diffraction noise. Solid curves were calculated by Eq. (3) with ε = 10% for PPLN (a) and ε = 1.7% for the reference (b).

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4. Results and analysis

The far-field diffraction pattern from our PPLN sample is shown in Fig. 2 in comparison with that from the reference grating. The graphs were normalized to the first order intensity I₁. For PPLN, the measured diffraction noise was I_n/I₁ = 1.2 × 10⁻⁴, from which we could estimate the STD using Eq. (4), ε = 10% of the grating period. The average duty ratio $\overline{R}$ could be estimated directly from the microscopic picture or from the ratio of the intensities of the two orders, I₂/I₁ [10]. The two methods gave $\overline{R}$ = 0.46 and 0.41, respectively, for the PPLN. This difference puts 0.3% additional uncertainty in the estimation of ε. For the reference grating, $\overline{R}$ = 0.49 and 0.45 from the direct microscopic observation and the measured I₂/I₁ value, respectively, and a small value of ε = 1.7% was obtained from the measured value of I_n/I₁ = 4.1 × 10⁻⁶.

In the above evaluation using Eq. (4), we took the noise intensity averaged around X = 1.5 (midpoint between the first and the second orders). However, the choice of this specific spatial frequency is somewhat arbitrary, in the sense that the noise intensity is not minimal at this point, but continues to decrease slightly with spatial frequency. Taking the noise intensity at the real minimum would not give more valuable data, because small contributions from the tails of the strong orders cannot be separated anyway. These contributions can be significantly reduced by a Gaussian beam input that creates smaller tails than the rect-beam [13].

A broad shoulder is observed for PPLN in the spatial frequency range of 1< X <1.4, and is more significant than that for the reference grating. If we fit the noise data in this range instead of the noise value at X = 1.5, the STD would be 14.5%. Although further investigation is necessary to understand the sources of the additional noise, one of the causes could be the departure of the real PPLN grating from our simple one-dimensional model defined by Eq. (1). Microscopic observation revealed that the reference grating is almost uniform in the y-direction perpendicular to the grating vector, where the one-dimensional model can be ideally applied. However, the PPLN grating had disorder not only in the x-direction, but also in the y-direction. Direct measurement of the domain wall width was carried out along a single horizontal line parallel to the x-axis, while the diffraction measurement used a natural laser beam width of ~1 mm in the y-dimension, which contains additional disorder, and could produce more noise in some spatial frequency region. The shoulder could be reduced if one minimizes the y-dimension of the illuminated area by focusing the input beam in this direction.

In order to verify the validity of the suggested diffraction method, we statistically analyzed the microscopic image of the PPLN grating. Figure 3 shows the histograms representing the statistics of the poled and unpoled domain widths in the 10 mm-long grating. For the poled domains [Fig. 3(a)], the average domain width was 9.8 μm, and the STD was 2.0 μm (ε = 9.4%), supporting the results obtained from the diffraction experiment. For the unpoled domains [Fig. 3(b)], the average domain width was 11.4 μm, and the STD was the same, as expected. The same analysis has been carried out on the reference grating. We obtained a relative STD of 1.5%, which also confirms the value of 1.7% obtained by the diffraction method within the measurement uncertainty. The STD always tends to be overestimated due to the finite slit width used in the measurement of the first order peak intensity. More accurate STD values could have been obtained by using a narrower slit or a long focal length lens for Fourier transform.

 

Fig. 3 Histograms showing statistics of the widths of poled (a) and unpoled (b) domains with corresponding Gaussian distributions.

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The statistical RDE information obtained by the above diffraction noise method gives the equivalent information that could be acquired by the tunable SHG experiment, where the equivalence has been verified in [10]. Furthermore, we can predict the parasitic noise in any frequency conversion process based on the estimated STD, as long as the pump wave depletion is not significant.

The diffraction noise measurement provides a versatile method for quantifying the poling quality of QPM devices, as long as the surface-etched structure projects the (averaged) ferroelectric domain structure. This condition may not be satisfied if the domain boundaries are not parallel to the ferroelectric axis (“slant domains”). In this case, the domain widths vary along the axis, resulting in a distributed duty ratio. Then, the diffraction pattern would not give the same information as in the SHG spectrum, because SHG would average the effects within the beam area. However, the RDE information obtained on the surface can still provide an essential figure for the poling quality for the slant domains in the crystal plate.

Another important point in the diffraction experiment is that surface etching is not required for PPLNs made of congruent LiNbO₃. Such PPLNs showed a diffraction pattern due to a small refractive index modulation between the poled and unpoled domains even without etching, resulting in the same quality estimation [10]. The index contrast was attributed to the stress field formed after the domain inversion. Further investigations are needed for other periodically poled ferroelectric crystals.

Finally, we should mention that a photodetector was scanned along the diffraction pattern in order to ensure linearity in this experiment. However, a CCD-camera with good linearity (or carefully calibrated CCD) can be used to obtain the diffraction pattern without scanning, shortening the measurement time dramatically.

5. Conclusion

We proposed diffraction noise measurement to evaluate the poling quality of PPLN. With this method, we quantified the statistical information about the duty-cycle error by analyzing the diffraction pattern. Our proposed method can provide a fast, low-cost, but accurate tool for evaluating the microscopic quality, and predicting the frequency conversion performance of QPM devices. Although random duty-cycle errors are inherent in the fabrication of ferroelectric QPM devices, our method can provide fast feedback to the poling process facilitating efficient quality control, for both users and manufacturers.